Question: To answer each question, use the function \(t\left( r \right)=\frac{d}{r}\), where t is the time in hours, d is the distance in miles, and r is the rate in miles per hour.

a. Sydney drives 10 mi at a certain rate and then drives 20 mi at a rate 5 mi/h faster than the initial rate. Write expressions for the time along each part of the trip. Add these times to write an equation for the total time in terms of the initial rate, \({{t}_{total}}\left( r \right)\).

b. Determine the reasonable domain and range and describe any discontinuities of \({{t}_{total}}\left( r \right)\) . Graph \({{t}_{total}}\left( r \right)\) on your graphing calculator.

c. At what rate, to the nearest mi/h, must Sydney drive if the entire 30 mi must be covered in about 45 min? Find the answer using the graph and using algebraic methods.

d. How long will Sydney take to drive the entire 30 mi if the car’s initial rate varies between 10 mi/h and 20 mi/h? Use the graph and algebraic methods to find the answer.